https://file.aiquickdraw.com/mathbot/file/17268426298798sx8i6ef.jpg

The inequality in the image is:

\[
0 \leq \frac{3x}{2} - 6 < 9
\]

Let's solve it step by step:

1. Start by dealing with the inequality as two separate parts:
   - \( 0 \leq \frac{3x}{2} - 6 \)
   - \( \frac{3x}{2} - 6 < 9 \)

### Solving the first part:
\[
0 \leq \frac{3x}{2} - 6
\]
Add 6 to both sides:
\[
6 \leq \frac{3x}{2}
\]
Multiply both sides by 2:
\[
12 \leq 3x
\]
Now, divide by 3:
\[
4 \leq x
\]
So, \( x \geq 4 \).

### Solving the second part:
\[
\frac{3x}{2} - 6 < 9
\]
Add 6 to both sides:
\[
\frac{3x}{2} < 15
\]
Multiply both sides by 2:
\[
3x < 30
\]
Now, divide by 3:
\[
x < 10
\]

### Final Solution:
Combining both results, we have:
\[
4 \leq x < 10
\]

So, the solution is \( x \in [4, 10) \).

Would you like further clarification or details?

Here are five related questions to expand on this concept:
1. How would you solve a compound inequality involving multiplication by a negative number?
2. What is the difference between solving inequalities and solving equations?
3. How can you graph the solution \( 4 \leq x < 10 \) on a number line?
4. What happens if you divide both sides of an inequality by a negative number?
5. How do you solve inequalities that contain absolute values?

**Tip:** When solving inequalities, always be careful when multiplying or dividing by negative numbers—they reverse the inequality sign!

Learn how to solve the linear inequality 0 ≤ (3x / 2) - 6 < 9 with a detailed step-by-step solution. This problem covers key algebraic concepts, including solving inequalities, linear transformations, and properties of inequalities. Suitable for students in Grades 8-10.

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